Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $q = \dfrac{2p(2p - 3)}{4p} \div \dfrac{20p - 30}{5} $
Explanation: Dividing by an expression is the same as multiplying by its inverse. $q = \dfrac{2p(2p - 3)}{4p} \times \dfrac{5}{20p - 30} $ When multiplying fractions, we multiply the numerators and the denominators. $q = \dfrac{ 2p(2p - 3) \times 5 } { 4p \times (20p - 30) } $ $ q = \dfrac {5 \times 2p(2p - 3)} {4p \times 10(2p - 3)} $ $ q = \dfrac{10p(2p - 3)}{40p(2p - 3)} $ We can cancel the $2p - 3$ so long as $2p - 3 \neq 0$ Therefore $p \neq \dfrac{3}{2}$ $q = \dfrac{10p \cancel{(2p - 3})}{40p \cancel{(2p - 3)}} = \dfrac{10p}{40p} = \dfrac{1}{4} $